The Aleksandrov Problem in Non-Archimedean 2-Fuzzy 2-Normed Spaces ()
1. Introduction
Let
be two metric spaces. For a mapping
, for all
, if f satisfies,
where
denote the metrics in the spaces
, then f is called an isometry. It means that for some fixed number
, assume that f preserves distance p; i.e., for all
in X, if
, we can get
. Then we say p is a conservative distance for the mapping f. Whether there exists a single conservative distance for some f such that f is an isometry from X to Y, is the basic issue of conservative distances. It is called the Aleksandrov problem.
Theorem 1.1. ( [1] ) Let
be two real normed linear spaces (or NLS) with
,
and Y is strictly convex, assume that a fixed real number
and that a fixed integer
. Finally, if
is a mapping satisfies
1)
2)
for all
. Then f is an affine isometry. we can call Benz’s theorem.
We can see some results about the Aleksandrov problem in different spaces in [2] - [10] . A natural question is that: Whether the Aleksandrov problem can be proved in non-Archimedean 2-fuzzy 2-normed spaces under some conditions. So in this article, we will give the definition of non-Archimedean 2-fuzzy 2-normed spaces according to [11] [12] [13] [14] , then by applying the Benz’s theorem to fix the value of p and N to solve problems.
If a function from a field K to
satisfies
(T1)
;
(T2)
;
(T3)
.
for all
, then the field K is called a non-Archimedean field.
We can know
,
for all
from the above definition. An example of a non-Archimedean valuation (or NAV) is the function
taking
and others into 1.
In 1897, Hence in [15] found that p-adic numbers play a vital role in the complex analysis, the norm derived from p-adic numbers is the non-Archimedean norm, the analysis of the non-Archimedean has important applications in physics.
Definition 1.2. Let X be a vector space and dim
. A function
is called non-Archimedean 2-norm, if and only if it satisfies
(T1)
,
iff
are linearly dependent;
(T2)
;
(T3)
;
(T4)
for all
. Then
is called non-Archimedean 2-normed space over the field K.
Definition 1.3. An NAV
in a linear space X over a field K. A function
is said to be a non-Archimedean fuzzy norm on X, if and only if for all
and
,
(F1)
with
,
(F2)
iff
for all
,
(F3)
, for
and
,
(F4)
,
(F5)
is a nondecreasing function of
and
.
Then
is known as a non-Archimedean fuzzy normed space (or F-NANS).
Theorem 1.4. Let
be an F-NANS. Assume the condition that:
(F6)
for all
.
Define
. We call these α-norms on X or the fuzzy norm on X.
Proof: 1) Let
, it implies that
, then for all
,
,
, so
;
Conversely, assume that
, by (F2),
for all
, then
for all
, so
.
2) By (F3), if
, then
Let
, then
If
, then
3) We have
Example 1.5. Let
be a non-Archimedean normed space. Define
for all
, Then
is a F-NANS.
Definition 1.6. Let Z be any non-empty set and
be the set of all fuzzy sets on Z. For
and
, define
and
Definition 1.7. A non-Archimedean fuzzy linear space
over the field K, we define the addition and scalar multiplication operation of X as following:
,
, if for every
, we have a related non-negative real numebr,
is the fuzzy norm of
in such that
(T1)
;
(T2)
;
(T3)
;
(T4)
for all
.
for every
, then we say that X is an F-NANS.
Definition 1.8. Let X be a non-empty non-Archimedean field set,
be the set of all fuzzy sets on X. If
, then
. Clearly,
, so
is a bounded function. Let
, then
is a non-Archimedean linear space over the field K and the addition, scalar multiplication are defined as follows
and
If for every
, there is a related non-negative real number
called the norm of f in such that for all
(T1)
iff
. For
(T2)
. For
(T3)
. For
Then the linear space
is a non-Archimedean normed space.
Definition 1.9. ( [4] ) A 2-fuzzy set on X is a fuzzy set on
.
Definition 1.10. A NAV
in a linear space
over a field K. If a function
is a non-Archimedean 2-fuzzy 2-norm on X (or a fuzzy 2-norm on
), iff for all
,
,
(F1)
for
;
(F2)
iff
are linearly dependent for all
;
(F3)
;
(F4)
, for
and
;
(F5)
;
(F6)
is a nondecreasing function of R and
;
Then
is called a non-Archimedean fuzzy 2-normed space (or FNA-2) or
is a non-Archimedean 2-fuzzy 2-normed space.
Theorem 1.11. Let
be an FNA-2. Suppose the condition that:
(F7)
for all
and
are linearly dependent.
Define
. We call these α-2-norms on
or the 2-fuzzy 2-norm on X.
Proof: It is similar to the proof of Theorem 1.4.
2. Main Result
From now on, if we have no other explanation, let
,
.
,
Definition 2.1. Let
be two FNA-2 and a mapping
. If for all
and
, we have
then
is called 2-isometry.
Definition 2.2. For a mapping
and
1) If
, then
, we say
satisfies the area one preserving property (AOPP).
2) If
, then
, we say
satisfies the area n for each n (AnPP).
Definition 2.3. We say a mapping
preserves collinear, if
mutually disjoint elements of
, then exist some real number t we have
Next, we denote
.
Lemma 2.4. Let
and
be two FNA-2. If
, a mapping
satisfies
and AOPP, then we can get
where
.
Proof: 1) Firstly, we prove that f preserves collinear. We assume that
, according to
, we get
then
and
are linearly dependent. So we obtain that
preserves collinear.
2) Secondly, we prove that when
, we can get
.
If
Let
, then
, so
Since
according to
, we have
Since f preserves collinear, so there exists a real number s such that
and
So, we get
This contradicts with
.
Lemma 2.5. Let
and
be two FNA-2. If a mapping
satisfies AOPP and preserves collinear, then
1)
is an injective;
2) if
, then
and
with
.
Proof: 1) We prove
is injective. Let
, since dim
, there exists an element
such that
are linearly independent. Hence
.
Let
, then
, and
satisfies AOPP, so
we can see
. So the mapping
is injective.
2) Let
mutually disjoint elements of
and
, so
. Since
is injective and preserves collinear, there exist
such that
Since dim
, there exist an element
such that
. Let
, then
and
So,
Since
, we get
and
According to the mapping
is injective, so
, and
Let
, so we have
Therefore
and
So
is additive.
From the lemma 2.4, we know that if
, then
satisfies 2-isometry.
so
and
is linearly dependent i.e.
.
Next we assume
,
and
Thus
, if
, then
, but
so
. It contradicts with
. Thus
.
Lemma 2.6. Let
and
be FNA-2. If
, a mapping
satisfies
and AOPP, then we can get for all
, we can get
.
Proof: From lemma 2.4, we know
preserves collinear.
For any
, there exist two numbers
such that
.
So,
and
By lemma 2.5, we have
Thus
Lemma 2.7. Let
and
be two FNA-2. If a mapping
satisfies AOPP and
for all
with
, then
satisfies AnPP.
Proof: Let
and
. Let
and
So,
We know
preserves collinear. So there exist a number
such that
Therefore
Then we have
. By lemma 2.5,
, so
.
In the same way, we can get
Hence
Therefore
Theorem 2.8. Let
and
be two FNA-2. If a mapping
satisfies AOPP and
for all
with
, then
is 2-isometry.
Proof: Since lemma 2.4, we just need to prove that
with
.
We can assume that when
, for all
, we have
. and there exist a number
such that
Let
, then
and
Since
preserves collinear, there exist a number
such that
Since 2),
which is contradiction, so
Therefore, we get
with
. Hence
for all
.